Evaluate the Summation sum from k=3 to 13 of 12k-62
The question is asking to perform the operation of summation on a given algebraic expression, specifically 12k-62. The summation is to be carried out for integer values of k starting at 3 and ending at 13. Essentially, it means to calculate the result of the expression 12k-62 for every integer value of k from 3 to 13, inclusive, and then find the total sum of these individual results. This is a mathematical problem involving arithmetic and algebraic manipulations.
$\sum_{k = 3}^{13} 12 k - 62$
Answer
Solution:
Step 1:
Decompose the original summation to reset the lower limit of $k$ to $1$.
$$\sum_{k = 3}^{13} (12k - 62) = \sum_{k = 1}^{13} (12k - 62) - \sum_{k = 1}^{2} (12k - 62)$$
Step 2:
Compute the summation from $k = 1$ to $k = 13$ for the expression $12k - 62$.
Step 2.1:
Break down the summation into two parts to apply standard summation formulas.
$$\sum_{k = 1}^{13} (12k - 62) = 12\sum_{k = 1}^{13} k - 62\sum_{k = 1}^{13} 1$$
Step 2.2:
Calculate the first part, $12\sum_{k = 1}^{13} k$.
Step 2.2.1:
Use the arithmetic series formula:
$$\sum_{k = 1}^{n} k = \frac{n(n + 1)}{2}$$
Step 2.2.2:
Insert the upper limit and multiply by the coefficient.
$$12 \cdot \frac{13 \cdot (13 + 1)}{2}$$
Step 2.2.3:
Simplify the expression.
Step 2.2.3.1:
Perform the addition inside the parentheses.
$$12 \cdot \frac{13 \cdot 14}{2}$$
Step 2.2.3.2:
Reduce the fraction by eliminating common factors.
Step 2.2.3.2.1:
Extract the factor of $2$ from $12$.
$$2 \cdot 6 \cdot \frac{182}{2}$$
Step 2.2.3.2.2:
Cancel out the $2$s.
$$\cancel{2} \cdot 6 \cdot \frac{182}{\cancel{2}}$$
Step 2.2.3.2.3:
Rewrite the simplified expression.
$$6 \cdot 182$$
Step 2.2.3.3:
Multiply $6$ by $182$ to get the result.
$$1092$$
Step 2.3:
Compute the second part, $-62\sum_{k = 1}^{13} 1$.
Step 2.3.1:
Apply the formula for the sum of a constant term.
$$\sum_{k = 1}^{n} c = c \cdot n$$
Step 2.3.2:
Substitute the constant and the upper limit.
$$-62 \cdot 13$$
Step 2.3.3:
Carry out the multiplication.
$$-806$$
Step 2.4:
Combine the results of the two summations.
$$1092 - 806$$
Step 2.5:
Subtract $806$ from $1092$ to get the final value.
$$286$$
Step 3:
Determine the summation from $k = 1$ to $k = 2$ for the expression $12k - 62$.
Step 3.1:
Write out the terms of the series for each $k$ value.
$$12 \cdot 1 - 62 + 12 \cdot 2 - 62$$
Step 3.2:
Simplify the series.
Step 3.2.1:
Multiply $12$ by $1$ and $2$ respectively.
$$12 - 62 + 24 - 62$$
Step 3.2.2:
Subtract $62$ from each product.
$$-50 + 24 - 62$$
Step 3.2.3:
Combine the results.
$$-50 - 38$$
Step 3.2.4:
Calculate the sum.
$$-88$$
Step 4:
Substitute the computed summations back into the original equation.
$$286 - 88$$
Step 5:
Add $286$ and $-88$ to get the final answer.
$$198$$
Knowledge Notes:
The problem involves evaluating a summation of a linear expression with respect to an integer variable $k$. The process requires knowledge of the following concepts:
Summation Notation: Understanding how to read and interpret the sigma notation, which is a compact way to represent the sum of a sequence of terms.
Arithmetic Series: Recognizing that the sum of the first $n$ positive integers is given by the formula $\sum_{k = 1}^{n} k = \frac{n(n + 1)}{2}$.
Sum of a Constant: Knowing that the sum of a constant $c$ over $n$ terms is simply $c \cdot n$.
Linearity of Summation: Utilizing the property that allows the summation to be split over addition or subtraction and factoring out constants, i.e., $\sum_{k = a}^{b} (c \cdot f(k) \pm d) = c \cdot \sum_{k = a}^{b} f(k) \pm d \cdot (b - a + 1)$.
Algebraic Manipulation: Applying basic algebraic operations such as addition, subtraction, multiplication, and simplification to evaluate expressions.
The solution process involves decomposing the original summation into two parts to reset the lower limit, applying summation formulas for arithmetic series and constants, and then simplifying and combining the results to find the final answer.
